First of all, the secrecy aspect of the genocide only fully applies to the first wave of mass extermination during the "Aktion Reinhardt". They did kill hundreds of thousands of people in KZs like Treblinka. As the stability of the regime was higher in these years, they managed to leave back almost no trace, which is astounding when you take the scale of the operation into consideration. The trickery the Nazis applied to make the Polish believe they were only deported to a "transit camp", is hauntingly precise. (For further reading: https://archive.ph/20140321212050/http://www.holocaust-history.org/german-trials/treblinka-urteil.shtml; it's in German, but maybe Google translate can help.)

Secondly, you have to look at the bigger picture. By 1944, the war was lost. The "final solution" was one of the last things Hitler and the Nazis actually could be "successful" at, because everything else was falling apart at a rapid rate. They firmly believed that mass murdering all Jews would somehow cure the world from all evil. (I'm paraphrasing here, but there are countless Hitler speeches in which he openly talks about exterminating all Jews.)

The holocaust in its final phase was NOT secret. There was a whole KZ economy system, there were thousands of SS men guarding the camps, there were mass deportations in all major areas the Nazis controlled. Hungary surely was special, as they were allied with Germany. Their leader, Miklós Horthy had deported thousands of Jews during the first years of his reign. He then stopped doing it, but forced Jewish laborers to fight against the Red Army. The real madness started, when Horthy's administration didn't cooperate during late 1943. Hitler wanted all hungarian jews to be deported, which led to "Operation Margarethe", which was closely linked to Adolf Eichmann. Around 400000 hungarian people were deported and murdered. Then Horthy finally drew the line due to pressure from the Allies and the Vatican, which led to the event you mentioned in your post.

So, why did they stop caring about secrecy? That's a complex question. The KZ system remained somewhat intact during the final period of the war. Hitler put a lot of effort into erasing evidence. A lot of things and people were burned during this stage of the war. In the end, the Holocaust was simply too big to be kept secret. There were too many eye-witnesses. The Nazi leadership was in full collapse during 1944 and 1945, the German economy was basically non-existant. Still, the Holocaust machine kept running up until January 1945, when all eastern camps were abandoned. The remaining Jews were either killed or sent on death marches.

To really understand the Nazi's actions, you have to remember that antisemitism was fundamental to the Nazi ideology. Jews were not humans. The war was lost, but the "bigger" fight was still on. They fully knew that it would be impossible to hide the Holocaust, which maybe is the main reason why they kept murdering Jews until the very last minute. There was nothing left to lose. It's the totality of their ideology that makes the Holocaust so unique. These were not political killings, these were industrialized exterminations for "the greater good".

You also have to remember that the Holocaust might have been Hitler's (and Himmler's) idea, but in the end, it was the doing of around 500000 Germans and collaborators who administered and executed the genocide.

Finally, it also has to be mentioned, that antisemitism was not a German "priviledge", e.g. the British administration spoke up against offering shelter to european jews due to "strategic reasons" in some occasions. I'm not saying they were antisemitic, but in general, the Jews were not as big a concern as you might think. This especially applies up until 1944. The Soviets also openly ignored the deportations.